Ebook: Postmodern Analysis

This is an introduction to advanced analysis at the beginning graduate level that blends a modern presentation with concrete examples and applications, in particular in the areas of calculus of variations and partial differential equations. The book does not strive for abstraction for its own sake, but tries rather to impart a working knowledge of the key methods of contemporary analysis, in particular those that are also relevant for application in physics. It provides a streamlined and quick introduction to the fundamental concepts of Banach space and Lebesgue integration theory and the basic notions of the calculus of variations, including Sobolev space theory.
The new edition contains additional material on the qualitative behavior of solutions of ordinary differential equations, some further details on Lp and Sobolev functions, partitions of unity and a brief introduction to abstract measure theory.

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This is an introduction to advanced analysis at the beginning graduate level that blends a modern presentation with concrete examples and applications, in particular in the areas of calculus of variations and partial differential equations. The book does not strive for abstraction for its own sake, but tries rather to impart a working knowledge of the key methods of contemporary analysis, in particular those that are also relevant for application in physics. It provides a streamlined and quick introduction to the fundamental concepts of Banach space and Lebesgue integration theory and the basic notions of the calculus of variations, including Sobolev space theory.
The new edition contains additional material on the qualitative behavior of solutions of ordinary differential equations, some further details on Lp and Sobolev functions, partitions of unity and a brief introduction to abstract measure theory.
Content:
Front Matter....Pages I-XVII
Front Matter....Pages 1-1
Prerequisites....Pages 3-12
Limits and Continuity of Functions....Pages 13-19
Differentiability....Pages 21-29
Characteristic Properties of Differentiable Functions. Differential Equations....Pages 31-42
The Banach Fixed Point Theorem. The Concept of Banach Space....Pages 43-46
Uniform Convergence. Interchangeability of Limiting Processes. Examples of Banach Spaces. The Theorem of Arzela-Ascoli....Pages 47-60
Integrals and Ordinary Differential Equations....Pages 61-73
Front Matter....Pages 75-75
Metric Spaces: Continuity, Topological Notions, Compact Sets....Pages 77-99
Front Matter....Pages 101-101
Differentiation in Banach Spaces....Pages 103-114
The Implicit Function Theorem. Applications....Pages 115-131
Front Matter....Pages 133-143
Preparations. Semicontinuous Functions....Pages 145-153
The Lebesgue Integral for Semicontinuous Functions. The Volume of Compact Sets....Pages 155-155
Lebesgue Integrable Functions and Sets....Pages 157-163
Null Functions and Null Sets. The Theorem of Fubini....Pages 165-182
The Convergence Theorems of Lebesgue Integration Theory....Pages 183-193
Measurable Functions and Sets. Jensen’s Inequality. The Theorem of Egorov....Pages 195-203
The Transformation Formula....Pages 205-215
Front Matter....Pages 217-227
Front Matter....Pages 229-237
Integration by Parts. Weak Derivatives. Sobolev Spaces....Pages 239-239
Front Matter....Pages 241-260
Hilbert Spaces. Weak Convergence....Pages 239-239
Variational Principles and Partial Differential Equations....Pages 261-282
Regularity of Weak Solutions....Pages 283-283
The Maximum Principle....Pages 285-294
The Eigenvalue Problem for the Laplace Operator....Pages 295-326
Back Matter....Pages 327-341
....Pages 343-353